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❖✉t❣r♦✇t❤s ♦❢ t❤❡ ❉▲▼❋ Pr♦❥❡❝t✿ P❛rt ✷✿ ◆■❙❚ ❉✐❣✐t❛❧ ❘❡♣♦s✐t♦r② ♦❢ ▼❛t❤❡♠❛t✐❝❛❧ ❋♦r♠✉❧❛❡

❍♦✇❛r❞ ❈♦❤❧✯✱ ▼❛r❥❡ ▼❝❈❧❛✐♥✯✱ ❇♦♥✐t❛ ❙❛✉♥❞❡rs✯✱ ▼♦r✐t③ ❙❝❤✉❜♦t③§

✯❆♣♣❧✐❡❞ ❛♥❞ ❈♦♠♣✉t❛t✐♦♥❛❧ ▼❛t❤❡♠❛t✐❝s ❉✐✈✐s✐♦♥✱ ◆■❙❚✱ ●❛✐t❤❡rs❜✉r❣✱ ▼❛r②❧❛♥❞✱ ❯✳❙✳❆✳ §❉❛t❛❜❛s❡ ❙②st❡♠s ❛♥❞ ■♥❢♦r♠❛t✐♦♥ ▼❛♥❛❣❡♠❡♥t ●r♦✉♣✱ ❚❡❝❤♥✐s❝❤❡ ❯♥✐✈❡rs✐tät ❇❡r❧✐♥

❈❤❛❧❧❡♥❣❡s ✐♥ ✷✶st ❈❡♥t✉r②✿ ❊①♣❡r✐♠❡♥t❛❧ ▼❛t❤❡♠❛t✐❝❛❧ ❈♦♠♣✉t❛t✐♦♥ ■♥st✐t✉t❡ ❢♦r ❈♦♠♣✉t❛t✐♦♥❛❧ ❛♥❞ ❊①♣❡r✐♠❡♥t❛❧ ❘❡s❡❛r❝❤ ✐♥ ▼❛t❤❡♠❛t✐❝s✱ Pr♦✈✐❞❡♥❝❡✱ ❘❤♦❞❡ ■s❧❛♥❞

❏✉❧② ✷✶✱ ✷✵✶✹

✭◆■❙❚✱ ❚❯ ❇❡r❧✐♥✮ ❉❘▼❋ ❏✉❧② ✷✶✱ ✷✵✶✹ ✶ ✴ ✶✸

❉✐❣✐t❛❧ ❘❡♣♦s✐t♦r② ♦❢ ▼❛t❤❡♠❛t✐❝❛❧ ❋♦r♠✉❧❛❡

❖♥❧✐♥❡ ❝♦♠♣❡♥❞✐✉♠ ♦❢ ♠❛t❤❡♠❛t✐❝❛❧ ❢♦r♠✉❧❛❡

♦rt❤♦❣♦♥❛❧ ♣♦❧②♥♦♠✐❛❧ ❛♥❞ s♣❡❝✐❛❧ ❢✉♥❝t✐♦♥ ❢♦r♠✉❧❛❡

❉❘▼❋ ❛tt❡♠♣ts t♦ ✉s❡ ❲❡❜ ✷✳✵ t❡❝❤♥♦❧♦❣✐❡s t♦ ♠♦✈❡ ❜❡②♦♥❞ t❤❡ st❛t✐❝ ♣r❡s❡♥t❛t✐♦♥ ♦❢ r❡❢❡r❡♥❝❡ ❞❛t❛ t♦ ❛ ♣❧❛t❢♦r♠ t❤❛t ❡♥❝♦✉r❛❣❡s ❝♦♠♠✉♥✐t② ✐♥t❡r❛❝t✐♦♥ ❛♥❞ ❝♦❧❧❛❜♦r❛t✐♦♥✳ ❉❘▼❋ ✉t✐❧✐③❡s ♦❢ ❉▲▼❋ ▲

❆

❚ ❊❳ ♠❛❝r♦s

t✐❡ s♣❡❝✐✜❝ ❝❤❛r❛❝t❡r s❡q✉❡♥❝❡s t♦ ✇❡❧❧✲❞❡✜♥❡❞ ♠❛t❤❡♠❛t✐❝❛❧ ♦❜❥❡❝ts✳ Pr♦✈✐❞❡s ❛♥ ✐♥t❡r♥❡t ❧✐♥❦ t♦ st❛♥❞❛r❞✱ ♣r❡❝✐s❡ ♦rt❤♦❣♦♥❛❧ ♣♦❧②♥♦♠✐❛❧ ❛♥❞ s♣❡❝✐❛❧ ❢✉♥❝t✐♦♥ ❞❡✜♥✐t✐♦♥s t❤r♦✉❣❤ t❤❡ ❉▲▼❋ ❛♥❞ ❉❘▼❋

❯s❡s ▼❡❞✐❛❲✐❦✐ ✇✐❦✐ s♦❢t✇❛r❡

▼❛t❤▼▲ s✉♣♣♦rt ▲

❆

❚ ❊❳▼▲ ▼❛t❤❏❛①

✭◆■❙❚✱ ❚❯ ❇❡r❧✐♥✮ ❉❘▼❋ ❏✉❧② ✷✶✱ ✷✵✶✹ ✷ ✴ ✶✸

❉❘▼❋ ❣♦❛❧s

❚❤❡ ❉❘▼❋ ✇✐❧❧ ❜❡ ❞❡s✐❣♥❡❞ ❢♦r ❛ ♠❛t❤❡♠❛t✐❝❛❧❧② ❧✐t❡r❛t❡ ❛✉❞✐❡♥❝❡ ❛♥❞ s❤♦✉❧❞✿

✶ ❢❛❝✐❧✐t❛t❡ ✐♥t❡r❛❝t✐♦♥ ❛♠♦♥❣ ❛ ❝♦♠♠✉♥✐t② ♦❢ ♠❛t❤❡♠❛t✐❝✐❛♥s ❛♥❞

s❝✐❡♥t✐sts ✐♥t❡r❡st❡❞ ✐♥ ❢♦r♠✉❧❛❡ ❞❛t❛ r❡❧❛t❡❞ t♦ ♦rt❤♦❣♦♥❛❧ ♣♦❧②♥♦♠✐❛❧s ❛♥❞ s♣❡❝✐❛❧ ❢✉♥❝t✐♦♥s ✭❖P❙❋✮❀

✷ ❜❡ ❡①♣❛♥❞❛❜❧❡✱ ❛❧❧♦✇✐♥❣ t❤❡ ✐♥♣✉t ♦❢ ♥❡✇ ❢♦r♠✉❧❛❡❀ ✸ ❜❡ ❛❝❝❡ss✐❜❧❡ ❛s ❛ st❛♥❞❛❧♦♥❡ r❡s♦✉r❝❡❀ ✹ ❤❛✈❡ ❛ ✉s❡r ❢r✐❡♥❞❧②✱ ❝♦♥s✐st❡♥t✱ ❛♥❞ ❤②♣❡r❧✐♥❦❛❜❧❡ ✈✐❡✇♣♦✐♥t ❛♥❞

❛✉t❤♦r✐♥❣ ♣❡rs♣❡❝t✐✈❡❀ ❛♥❞

✺ ❝♦♥t❛✐♥ ❡❛s✐❧② s❡❛r❝❤❛❜❧❡ ♠❛t❤❡♠❛t✐❝s ❛♥❞ t❛❦❡ ❛❞✈❛♥t❛❣❡ ♦❢

♠♦❞❡r♥ ▼❛t❤▼▲ t♦♦❧s ❢♦r ❡❛s② t♦ r❡❛❞✱ s❝❛❧❛❜❧② r❡♥❞❡r❡❞ ♠❛t❤❡♠❛t✐❝s✳

✭◆■❙❚✱ ❚❯ ❇❡r❧✐♥✮ ❉❘▼❋ ❏✉❧② ✷✶✱ ✷✵✶✹ ✸ ✴ ✶✸

❉❘▼❋ ❙❡❡❞✐♥❣ Pr♦❥❡❝ts ▼❛t❤ ❖❈❘✱ ▼❛❝r♦ r❡♣❧❛❝❡♠❡♥t✱ ❛♥❞ ❲✐❦✐t❡①t ❣❡♥❡r❛t✐♦♥✱ t♦ ✐♠♣❧❡♠❡♥t ♣r❡✲❡①✐st✐♥❣ ❜♦♦❦ ❝♦♠♣❡♥❞✐❛

▼❛t❤❡♠❛t✐❝❛❧ ❖♣t✐❝❛❧ ❈❤❛r❛❝t❡r ❘❡❝♦❣♥✐t✐♦♥ ♣r♦❥❡❝t

❇❛t❡♠❛♥ ♠❛♥✉s❝r✐♣t ♣r♦❥❡❝t✿ ❍✐❣❤❡r ❚r❛♥s❝❡♥❞❡♥t❛❧ ❋✉♥❝t✐♦♥s✱ ❚❛❜❧❡s ♦❢ ■♥t❡❣r❛❧ ❚r❛♥s❢♦r♠s ❇②r❞ ✫ ❋r✐❡❞♠❛♥✬s ❍❛♥❞❜♦♦❦ ♦❢ ❊❧❧✐♣t✐❝ ■♥t❡❣r❛❧s ❢♦r ❊♥❣✐♥❡❡rs ❛♥❞ ❙❝✐❡♥t✐sts

❉▲▼❋ ▲

❆

❚ ❊❳ ♠❛❝r♦ r❡♣❧❛❝❡♠❡♥t ♣r♦❥❡❝t

❍②♣❡r❣❡♦♠❡tr✐❝ ❖rt❤♦❣♦♥❛❧ P♦❧②♥♦♠✐❛❧s ❛♥❞ ❚❤❡✐r q✲❆♥❛❧♦❣✉❡s ✕ ❑▲❙ ❑▲❙ ❛❞❞❡♥❞✉♠ ❜② ❚♦♠ ❑♦♦r♥✇✐♥❞❡r ❢✉t✉r❡❄✿ ❆♥❞r❡✇s✱ ❆s❦❡② ✫ ❘♦② ✿ ❙♣❡❝✐❛❧ ❋✉♥❝t✐♦♥s ❢✉t✉r❡❄✿ ■s♠❛✐❧ ✿ ❈❧❛ss✐❝❛❧ ❛♥❞ ◗✉❛♥t✉♠ ❖rt❤♦❣♦♥❛❧ P♦❧②♥♦♠✐❛❧s ✐♥ ❖♥❡ ❱❛r✐❛❜❧❡ ❢✉t✉r❡❄✿ ❡t❝✳

❲✐❦✐t❡①t ❣❡♥❡r❛t✐♦♥ ♣r♦❥❡❝t

◆■❙❚ ❉✐❣✐t❛❧ ▲✐❜r❛r② ♦❢ ▼❛t❤❡♠❛t✐❝❛❧ ❋✉♥❝t✐♦♥s ✭❝❤✳ ✷✺✮ ✿ ✶✼✵ ❢♦r♠✉❧❛s

✭◆■❙❚✱ ❚❯ ❇❡r❧✐♥✮ ❉❘▼❋ ❏✉❧② ✷✶✱ ✷✵✶✹ ✹ ✴ ✶✸

❉❘▼❋ ❩❡t❛ ❛♥❞ ❘❡❧❛t❡❞ ❋✉♥❝t✐♦♥s P❛❣❡ ✳✳✳ ❉▲▼❋ ❲✐❦✐t❡①t

· · · · · · · · · · · · · · · · · ·

✭◆■❙❚✱ ❚❯ ❇❡r❧✐♥✮ ❉❘▼❋ ❏✉❧② ✷✶✱ ✷✵✶✹ ✺ ✴ ✶✸

❉▲▼❋ ♠❛❝r♦s ♣r♦✈✐❞❡ s❡♠❛♥t✐❝ ❝♦♥t❡♥t ✐♥ ❢♦r♠✉❧❛s

❉▲▼❋ ❖P❙❋ ▼❛❝r♦s ✈✐❛ ▲

❆

❚ ❊❳▼▲✲s❡r✈❡r

✺✹✻ s❡♠❛♥t✐❝ ❉▲▼❋ ▲

❆❚

❊❳ ❖P❙❋ ♠❛❝r♦s ❛❞❞✐t✐♦♥❛❧ ✸✽ s❡♠❛♥t✐❝ ▲

❆

❚ ❊❳ ♠❛❝r♦s

❖❜❥❡❝ts✿ ❭s✉♠✱❭✐♥t✱❭❞❡r✐✈④❢⑥④①⑥✱❭q❞❡r✐✈❬♥❪④q⑥❅④③⑥ ❈♦♥st❛♥ts✿ ❭❡①♣❡✱❭✐✉♥✐t✱❭❝♣✐✱❭❊✉❧❡r❈♦♥st❛♥t ❙♣❡❝✐❛❧ ❋✉♥❝t✐♦♥s ❛♥❞ ❖rt❤♦❣♦♥❛❧ P♦❧②♥♦♠✐❛❧s

Γ(z) ❭❊✉❧❡r●❛♠♠❛❅④③⑥ ❤tt♣✿✴✴❞❧♠❢✳♥✐st✳❣♦✈✴✺✳✸✵★❊✶ Jν(z) ❭❇❡ss❡❧❏④❭♥✉⑥❅④③⑥ ❤tt♣✿✴✴❞❧♠❢✳♥✐st✳❣♦✈✴✶✵✳✷★❊✷ Qµ

ν(z)

❭▲❡❣❡♥❞r❡◗❬❭♠✉❪④❭♥✉⑥❅④③⑥✿ ❤tt♣✿✴✴❞❧♠❢✳♥✐st✳❣♦✈✴✶✹✳✸★❊✼ P (α,β)

n

(x) ❭❏❛❝♦❜✐P④❭❛❧♣❤❛⑥④❭❜❡t❛⑥④♥⑥❅④①⑥ ❤tt♣✿✴✴❞❧♠❢✳♥✐st✳❣♦✈✴✶✽✳✸★❚✶✳t✶✳r✸

✭◆■❙❚✱ ❚❯ ❇❡r❧✐♥✮ ❉❘▼❋ ❏✉❧② ✷✶✱ ✷✵✶✹ ✻ ✴ ✶✸

▼❛t❤❡♠❛t✐❝❛❧ ❋♦r♠✉❧❛❡

❲❤❡r❡❛s ❲✐❦✐♣❡❞✐❛ ❛♥❞ ♦t❤❡r ✇❡❜ ❛✉t❤♦r✐♥❣ t♦♦❧s ♠❛♥✐❢❡st ♥♦t✐♦♥s ♦r ❞❡s❝r✐♣t✐♦♥s ❛s ✜rst ❝❧❛ss ♦❜❥❡❝ts✱ t❤❡ ❉❘▼❋ ❞♦❡s t❤❛t ✇✐t❤ ♠❛t❤❡♠❛t✐❝❛❧ ❢♦r♠✉❧❛❡✳ ❉❘▼❋ ♣r♦✈✐❞❡s ❢♦r ❡❛❝❤ ❢♦r♠✉❧❛✱ ❛ ❢♦r♠✉❧❛ ❤♦♠❡ ♣❛❣❡✿

✶ ❘❡♥❞❡r❡❞ ❞❡s❝r✐♣t✐♦♥ ♦❢ t❤❡ ❢♦r♠✉❧❛ ✭r❡q✉✐r❡❞✮❀ ✷ ❈♦♥str❛✐♥ts t❤❡ ❢♦r♠✉❧❛ ♠✉st ♦❜❡② ✸ ❙✉❜st✐t✉t✐♦♥s r❡q✉✐r❡❞ t♦ ✉♥❞❡rst❛♥❞ ❢♦r♠✉❧❛❀ ✹ ❇✐❜❧✐♦❣r❛♣❤✐❝ ❝✐t❛t✐♦♥ ✭r❡q✉✐r❡❞✮❀ ✺ ❖♣❡♥ s❡❝t✐♦♥ ❢♦r ♣r♦♦❢s ✭r❡q✉✐r❡❞✮ ✕ ❉▲▼❋❀ ✻ ▲✐st ♦❢ s②♠❜♦❧s ❛♥❞ ❧✐♥❦s t♦ ❞❡✜♥✐t✐♦♥s ✭r❡q✉✐r❡❞✮ ✕ ❉▲▼❋ ♠❛❝r♦s❀ ✼ ❖♣❡♥ s❡❝t✐♦♥ ❢♦r ♥♦t❡s ✕ ❝♦♥♥❡❝t✐♦♥s ❜❡t✇❡❡♥ ❢♦r♠✉❧❛s❀ ❛♥❞ ✽ ❖♣❡♥ s❡❝t✐♦♥ ❢♦r ❡①t❡r♥❛❧ ❧✐♥❦s ✕ ❝♦♠♣✉t❡r ❣❡♥❡r❛t❡❞ ♣r♦♦❢s❀

✭◆■❙❚✱ ❚❯ ❇❡r❧✐♥✮ ❉❘▼❋ ❏✉❧② ✷✶✱ ✷✵✶✹ ✼ ✴ ✶✸

❙❛♠♣❧❡ ❢♦r♠✉❧❛ ❤♦♠❡ ♣❛❣❡

✭◆■❙❚✱ ❚❯ ❇❡r❧✐♥✮ ❉❘▼❋ ❏✉❧② ✷✶✱ ✷✵✶✹ ✽ ✴ ✶✸

❋✉rt❤❡r q✉❡st✐♦♥s

❍♦✇ ❞♦❡s ♦♥❡ ❢❛❝✐❧✐t❛t❡ ❡✛❡❝t✐✈❡ ❝♦♠♠✉♥✐t② ✐♥t❡r❛❝t✐♦♥ ✫ ❝♦♥tr✐❜✉t✐♦♥ ✇✐t❤ s✉❝❤ ❛ r❡s♦✉r❝❡❄

✐♠♣❧❡♠❡♥t ❛ ❤✐❣❤ ❞❡❣r❡❡ ♦❢ ❝♦♠♣✉t❡r ✈❡r✐✜❝❛t✐♦♥ ♦❢ ❝♦♠♠✉♥✐t② ✐♥♣✉t ❡♥s✉r❡ ❛ ❞❡❣r❡❡ ♦❢ ♠♦❞❡r❛t✐♦♥ ✐♥ t❤❡ ✇✐❦✐

❈❛♥ ♦♥❡ ❜✉✐❧❞ ❛ ♣✐❡❝❡ ♦❢ ✐♥t❡❧❧✐❣❡♥t s♦❢t✇❛r❡ ✇❤✐❝❤ ✐s ❛❜❧❡ t♦

s❝❛♥ ✐♥ ❜♦♦❦s❀ ♣r♦❞✉❝❡s ▲

❆

❚ ❊❳ s♦✉r❝❡❀ r❡♣❧❛❝❡s ❝♦♠♠❛♥❞s ❢♦r ❢✉♥❝t✐♦♥s ✐♥ t❤❡ s♦✉r❝❡ ✇✐t❤ s❡♠❛♥t✐❝ ♠❛❝r♦s❀ ❡①tr❛❝ts ❞❛t❛ ❢r♦♠ t❤❡ t❡①t ✭s✉❝❤ ❛s ❝♦♥str❛✐♥ts✮ ❛ss♦❝✐❛t❡s ❞❛t❛ ✇✐t❤ r❡❧❡✈❛♥t ❢♦r♠✉❧❛❡ ❛♥❞ r❡♠♦✈❡s t❡①t❀ ♣r♦❞✉❝❡s ❲✐❦✐t❡①t❀ ❛♥❞ ✉♣❧♦❛❞s ❲✐❦✐t❡①t t♦ ❛ ♣✉❜❧✐❝❧② ❛❝❝❡ss✐❜❧❡ ✇❡❜s✐t❡❄

❍♦✇ ❞♦❡s ♦♥❡ s❡❛r❝❤ t❤❡ r❡s✉❧t✐♥❣ ♠❛t❤❡♠❛t✐❝❛❧ ❞❛t❛❜❛s❡❄

✭◆■❙❚✱ ❚❯ ❇❡r❧✐♥✮ ❉❘▼❋ ❏✉❧② ✷✶✱ ✷✵✶✹ ✾ ✴ ✶✸

❖♥❣♦✐♥❣ ♣r♦❥❡❝ts t♦ ✐♥✈❡st✐❣❛t❡ t❤❡ ❛❜♦✈❡ q✉❡st✐♦♥s

▼❛❝r♦ r❡♣❧❛❝❡♠❡♥ts ❢r♦♠ ✇❡❧❧✲❝♦♥str✉❝t❡❞ ▲

❆❚

❊❳ s♦✉r❝❡ ❊①tr❛❝t✐♦♥ ♦❢ ♠❛t❤❡♠❛t✐❝❛❧ ❞❛t❛ ❢r♦♠ t❡①t ✭❦❡②✇♦r❞s✮ ❲✐❦✐t❡①t ❣❡♥❡r❛t✐♦♥ P♦rt✐♥❣ t❤❡ ❉▲▼❋ s❡❛r❝❤ ❡♥❣✐♥❡ ✐♥ ▼❡❞✐❛❲✐❦✐ ✭❉❘▼❋✮ ❖✉t♣✉t ♦❢ ❢♦r♠✉❧❛ ❞❛t❛ ❢r♦♠ r✐❣❤t✲❝❧✐❝❦❛❜❧❡ ♠❡♥✉s ✐♥ ❛ ✈❛r✐❡t② ♦❢ ❢♦r♠❛ts s♦ t❤❛t ❢♦r♠✉❧❛s ❝❛♥ ❜❡ ✉s❡❞ ❛♥❞ ❛❧s♦ ✈❡r✐✜❡❞

▲

❆❚

❊❳ ❡①♣❛♥❞❡❞ ▲

❆❚

❊❳ s❡♠❛♥t✐❝ ♣r❡s❡♥t❛t✐♦♥ ▼❛t❤▼▲ ❝♦♥t❡♥t ▼❛t❤▼▲ ▼❛t❤❡♠❛t✐❝❛ ▼❛♣❧❡ ❙❛❣❡

✭◆■❙❚✱ ❚❯ ❇❡r❧✐♥✮ ❉❘▼❋ ❏✉❧② ✷✶✱ ✷✵✶✹ ✶✵ ✴ ✶✸

❱✐rt✉❛❧ ♠❛❝❤✐♥❡s ❱✐rt✉❛❧ ▼❛❝❤✐♥❡ ■♥st❛♥❝❡s✿ ❳❙❊❉❊ ♣r♦❥❡❝t ✷ ❳❙❊❉❊ ❈❡♥t❖❙✿ ❞❡♠♦ ❛♥❞ ❞❡♣❧♦②♠❡♥t ✷ ❳❙❊❉❊ ❯❜✉♥t✉ s❡r✈❡r✿ ▲

❆

❚ ❊❳▼▲✱ ▼❛t❤♦✐❞ ❲✐❦✐♠❡❞✐❛ ❋♦✉♥❞❛t✐♦♥ ✕ ❲✐❦✐t❡❝❤ ✹ ❲▼❋ ❱❛❣r❛♥t ✐♥st❛♥❝❡s

✭◆■❙❚✱ ❚❯ ❇❡r❧✐♥✮ ❉❘▼❋ ❏✉❧② ✷✶✱ ✷✵✶✹ ✶✶ ✴ ✶✸

❆❝❦♥♦✇❧❡❞❣❡♠❡♥ts ▼♦r✐t③ ❙❝❤✉❜♦t③ ✭❚❯✲❇❡r❧✐♥✮✿ ▼❡❞✐❛❲✐❦✐ ▼❛t❤ ❇r✉❝❡ ▼✐❧❧❡r ✭◆■❙❚✮ ✿ ❉▲▼❋ ▼❛❝r♦s ❏❛♥❡❧❧❡ ❲✐❧❧✐❛♠s ✭❱❙❯✮ ✿ ✷✵✶✸ ❙❯❘❋ st✉❞❡♥t ❍✐❣❤ ❙❝❤♦♦❧ ❙t✉❞❡♥ts✿ ❏❛❦❡ ▼✐❣❞❛❧❧ ✕ ▼❛t❤❏❛① ♠❡♥✉ ❝✉st♦♠✐③❛t✐♦♥ ❈❤❡rr② ❩♦✉ ✕ s❡❡❞✐♥❣✴♠❛❝r♦ r❡♣❧❛❝❡♠❡♥t ❆❧❡① ❉❛♥♦✛ ✕ s❡❡❞✐♥❣✴♠❛❝r♦ r❡♣❧❛❝❡♠❡♥t ❆♠❜❡r ▲✐✉ ✕ ▼❛t❤❏❛① ♠❡♥✉ ❝✉st♦♠✐③❛t✐♦♥ ❏✐♠♠② ▲✐ ✕ ♠❛t❤❡♠❛t✐❝❛❧ s❡❛r❝❤

✭◆■❙❚✱ ❚❯ ❇❡r❧✐♥✮ ❉❘▼❋ ❏✉❧② ✷✶✱ ✷✵✶✹ ✶✷ ✴ ✶✸

P♦st❡r s❡ss✐♦♥ ✇✐t❤ ✇❡❜s✐t❡ ❞❡♠♦s ✿ ❲❡❞✳ ❡✈❡♥✐♥❣

NIST Digital Repository

• f Mathematical Formulae (DRMF)

Howard S. Cohl*, Marjorie A. McClain*, Bonita V. Saunders*, Moritz Schubotz§, Janelle C. Williams†

• A digital compendium of math formulae for orthogonal polynomials and special functions and

• Offshoot project of NIST Digital Library of Math Functions (DLMF) using MediaWiki and MediaWiki

• Use of DLMF semantic L

• DRMF treats formulae as first class objects, describing them in formula home pages which contain:

• Wikitext generation and semantic DLMF L

http://gw32.iu.xsede.org/index.php/Main_Page http://www.siam.org/meetings/opsfa13

• Percy Deift, Courant Institute of Mathematical Sciences, New York University, USA
• Charles F. Dunkl, University of Virginia, USA
• Olga Holtz, Technische Universität Berlin, Germany
• Mourad E.H. Ismail, University of Central Florida, USA
• Teresa E. Pérez Fernández, Universidad de Granada, Spain
• Sarah Post, University of Hawaii at Manoa, USA
• Nico Temme, Centrum Wiskunde & Informatica (CWI), The Netherlands
• Craig A. Tracy, University of California Davis, USA
• Lauren Williams, University of California Berkeley, USA
• Wadim Zudilin, The University of Newcastle, Australia
• Alexei Zhedanov, Donetsk Institute for Physics and Technology, Ukraine

• classical analysis
• approximation theory
• continued fractions
• potential theory
• q-calculus
• asymptotics
• Riemann-Hilbert problems
• random matrix theory
• superintegrability and supersymmetry
• integrable systems
• Painlevé equations
• orthogonal polynomials and special functions of several variables
• orthogonal polynomials associated with root systems
• spherical functions
• orthogonality on the complex plane
• multiple orthogonal polynomials
• Sobolev orthogonal polynomials
• stochastic processes

• science and industry
• handbooks
• numerical algorithms and tables
• symbolic computation
• combinatorics
• number theory
• theoretical physics
• probability theory and statistics

• Diego Dominici, State University of New York at New Paltz, USA
• Daniel W. Lozier, National Institute of Standards and Technology, USA

• Richard A. Askey, University of Wisconsin, USA
• Howard S. Cohl, National Institute of Standards and Technology, USA
• Kathy Driver, University of Cape Town, South Africa
• Tom H. Koornwinder, University of Amsterdam, The Netherlands
• Robert S. Maier, University of Arizona, USA
• Zeinab Mansour, King Saud University, Saudi Arabia
• Andrei Martinez-Finkelshtein, Universidad de Almeria, Almeria, Spain
• Willard Miller, University of Minnesota, USA
• Victor H. Moll, Tulane University, USA
• Adri Olde Daalhuis, The University of Edinburgh, UK
• Audrey Terras, University of California San Diego, USA
• Walter Van Assche, Katholieke Universiteit Leuven, Belgium
• Luc Vinet, University of Montreal, Canada

✭◆■❙❚✱ ❚❯ ❇❡r❧✐♥✮ ❉❘▼❋ ❏✉❧② ✷✶✱ ✷✵✶✹ ✶✸ ✴ ✶✸

July 21, 2014 ICERM Workshop, Providence 1

Outgrowths of the Digital Library of Mathematical Functions Project Part 1

DLMF Standard Reference Tables http://dlmftables.uantwerpen.be

Daniel Lozier

NIST, Gaithersburg, MD, USA

Outline

• Introduction

– DLMF – DLMF Standard Reference Tables – Algorithms and software

• Examples

– Table generation – Table comparison

• Acknowledgements

July 21, 2014 ICERM Workshop, Providence 2

http://dlmf.nist.gov

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July 21, 2014 ICERM Workshop, Providence 6

NIST Digital Library of Math Fcns

and

NIST Handbook of Math Fcns

compared to

Abramowitz and Stegun, 1964

July 21, 2014 ICERM Workshop, Providence 7

DLMF Std. Reference Tables Accurate function values on demand

• Keyed to http://dlmf.nist.gov
• Web input of

– Desired arguments – Desired precision

• Web output of

– Enclosures computed to 5 digits higher precision

• r one of 5 rounding modes at desired precision

July 21, 2014 ICERM Workshop, Providence 8

DLMF Std. Reference Tables Accuracy tests of function values from other sources

• Web input of

– Arguments and function values from an external source

• Web output of

– Digit-by-digit comparison – Erroneous digits shown in red – Relative error

July 21, 2014 ICERM Workshop, Providence 9

Algorithms and Software

• Special functions computed to arbitrary

precision and guaranteed accuracy

• Real variables, decimal or binary arithmetic
• Algorithms depend on detailed error analysis

with series and continued fractions

• For details see

– A. Cuyt et al, Handbook of Continued Fractions, Springer, 2008 – F. Backeljauw et al, Validated Evaluation of Special Mathematical Functions, Science of Computer Programming, v. 90A, 2014, pp. 2-20

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Example

table generation Bessel function , integer order, real argument

July 21, 2014 ICERM Workshop, Providence 11

) (x J n

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in base 10

July 21, 2014 ICERM Workshop, Providence 13

Example

July 21, 2014 ICERM Workshop, Providence 14

table comparison Matlab’s error function , real argument ) erf(x

July 21, 2014 ICERM Workshop, Providence 15

July 21, 2014 ICERM Workshop, Providence 16

Acknowledgements

• Primary NIST Collaborators

– Bruce Miller – Marje McClain – Bonita Saunders

• Primary Antwerp Collaborators

– Franky Backeljauw – Stefan Becuwe – Annie Cuyt

July 21, 2014 ICERM Workshop, Providence 29

Feedback

help us provide an excellent data service!!

• Digit display

– Do you like or dislike? – Do you prefer just relative errors?

• Extra digits and color

– Do you like or dislike?

• Please send ideas for

– Further development – Further application areas

July 21, 2014 ICERM Workshop, Providence 30